If X and Y are two sets such that `X uu Y` has 50 elements, X has 28 elements and Y has 32 elements, how many elements does `X nn Y` have?

Answer

To find the number of elements in the intersection of sets X and Y (denoted as \(X \cap Y\)), we can use the principle of inclusion-exclusion. The formula for the union of two sets is given by:

\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]

Where:
- \(n(X \cup Y)\) is the number of elements in the union of sets X and Y.
- \(n(X)\) is the number of elements in set X.
- \(n(Y)\) is the number of elements in set Y.
- \(n(X \cap Y)\) is the number of elements in the intersection of sets X and Y.

### Step-by-step Solution:

1. **Identify the given values**:
- \(n(X \cup Y) = 50\) (the total number of elements in the union of X and Y)
- \(n(X) = 28\) (the number of elements in set X)
- \(n(Y) = 32\) (the number of elements in set Y)

2. **Substitute the values into the formula**:
\[ 50 = 28 + 32 - n(X \cap Y) \]

3. **Calculate the sum of \(n(X)\) and \(n(Y)\)**:
\[ 28 + 32 = 60 \]
So, we can rewrite the equation as:
\[ 50 = 60 - n(X \cap Y) \]

4. **Rearrange the equation to solve for \(n(X \cap Y)\)**:
\[ n(X \cap Y) = 60 - 50 \]

5. **Perform the subtraction**:
\[ n(X \cap Y) = 10 \]

### Conclusion:
The number of elements in the intersection of sets X and Y is \(10\).

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